Problem 22 · 2016 Math Kangaroo
Stretch
Logic & Word Problems
magic-squaresum-constraint
Kirsten has written numbers into 5 of the 10 circles. She wants to write numbers into the remaining circles so that the sum of the three numbers along every side of the pentagon is always the same. Which number does she have to write into the circle marked X?
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Answer: D — 13
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Hint 1 of 3
Every side of the pentagon uses three circles and they all add to the same total.
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Hint 2 of 3
Find a side that already has two numbers filled in to pin down that common total.
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Hint 3 of 3
Then walk around the pentagon, filling each missing circle from the side total until you reach X.
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Approach: find the common side-total, then fill circles one at a time
- Call the common total of each side \(S\); a side that already shows two numbers tells you \(S\) once you know the third.
- Using the five given numbers and that fixed total \(S\), fill the empty circles one side at a time, each missing circle being \(S\) minus the two known circles on its side.
- Carrying this around to the marked circle gives \(X = 13\), choice (D).
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